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Theorem ordtypelem10 7847
Description: Lemma for ordtype 7852. Using ax-rep 4506, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1  |-  F  = recs ( G )
ordtypelem.2  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
ordtypelem.3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
ordtypelem.5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
ordtypelem.6  |-  O  = OrdIso
( R ,  A
)
ordtypelem.7  |-  ( ph  ->  R  We  A )
ordtypelem.8  |-  ( ph  ->  R Se  A )
Assertion
Ref Expression
ordtypelem10  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Distinct variable groups:    v, u, C    h, j, t, u, v, w, x, z, R    A, h, j, t, u, v, w, x, z    t, O, u, v, x    ph, t, x    h, F, j, t, u, v, w, x, z
Allowed substitution hints:    ph( z, w, v, u, h, j)    C( x, z, w, t, h, j)    T( x, z, w, v, u, t, h, j)    G( x, z, w, v, u, t, h, j)    O( z, w, h, j)

Proof of Theorem ordtypelem10
Dummy variables  b 
c  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3  |-  F  = recs ( G )
2 ordtypelem.2 . . 3  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . 3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . 3  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . 3  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . 3  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . 3  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem8 7845 . 2  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
91, 2, 3, 4, 5, 6, 7ordtypelem4 7841 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
10 frn 5668 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
12 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  A
)
136adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R  We  A
)
147adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R Se  A )
151, 2, 3, 4, 5, 13, 14ordtypelem8 7845 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
16 isof1o 6120 . . . . . . . . . . . . 13  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
17 f1of 5744 . . . . . . . . . . . . 13  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O --> ran  O )
1815, 16, 173syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O --> ran  O )
19 f1of1 5743 . . . . . . . . . . . . . 14  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O -1-1-> ran 
O )
2015, 16, 193syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O
-1-1-> ran  O )
21 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  A  /\  -.  b  e.  ran  O )  ->  b  e.  A )
22 seex 4786 . . . . . . . . . . . . . . 15  |-  ( ( R Se  A  /\  b  e.  A )  ->  { c  e.  A  |  c R b }  e.  _V )
237, 21, 22syl2an 477 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  { c  e.  A  |  c R b }  e.  _V )
2411adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  A
)
25 rexnal 2849 . . . . . . . . . . . . . . . . . . 19  |-  ( E. m  e.  dom  O  -.  ( O `  m
) R b  <->  -.  A. m  e.  dom  O ( O `
 m ) R b )
261, 2, 3, 4, 5, 6, 7ordtypelem7 7844 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( ( O `  m ) R b  \/  b  e.  ran  O ) )
2726ord 377 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( -.  ( O `
 m ) R b  ->  b  e.  ran  O ) )
2827rexlimdva 2941 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  A )  ->  ( E. m  e.  dom  O  -.  ( O `  m ) R b  ->  b  e.  ran  O ) )
2925, 28syl5bir 218 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  A. m  e.  dom  O ( O `  m
) R b  -> 
b  e.  ran  O
) )
3029con1d 124 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  A. m  e.  dom  O ( O `  m
) R b ) )
3130impr 619 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. m  e.  dom  O ( O `  m
) R b )
32 ffun 5664 . . . . . . . . . . . . . . . . . . . 20  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
339, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  Fun  O )
34 funfn 5550 . . . . . . . . . . . . . . . . . . 19  |-  ( Fun 
O  <->  O  Fn  dom  O )
3533, 34sylib 196 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  O  Fn  dom  O
)
3635adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Fn  dom  O )
37 breq1 4398 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( O `  m )  ->  (
c R b  <->  ( O `  m ) R b ) )
3837ralrn 5950 . . . . . . . . . . . . . . . . 17  |-  ( O  Fn  dom  O  -> 
( A. c  e. 
ran  O  c R
b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3936, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ( A. c  e.  ran  O  c R b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
4031, 39mpbird 232 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. c  e.  ran  O  c R b )
41 ssrab 3533 . . . . . . . . . . . . . . 15  |-  ( ran 
O  C_  { c  e.  A  |  c R b }  <->  ( ran  O 
C_  A  /\  A. c  e.  ran  O  c R b ) )
4224, 40, 41sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  { c  e.  A  |  c R b } )
4323, 42ssexd 4542 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  e.  _V )
44 f1dmex 6652 . . . . . . . . . . . . 13  |-  ( ( O : dom  O -1-1-> ran 
O  /\  ran  O  e. 
_V )  ->  dom  O  e.  _V )
4520, 43, 44syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  dom  O  e.  _V )
46 fex 6054 . . . . . . . . . . . 12  |-  ( ( O : dom  O --> ran  O  /\  dom  O  e.  _V )  ->  O  e.  _V )
4718, 45, 46syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  e.  _V )
481, 2, 3, 4, 5, 13, 14, 47ordtypelem9 7846 . . . . . . . . . 10  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
49 isof1o 6120 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  A
)  ->  O : dom  O -1-1-onto-> A )
50 f1ofo 5751 . . . . . . . . . 10  |-  ( O : dom  O -1-1-onto-> A  ->  O : dom  O -onto-> A
)
51 forn 5726 . . . . . . . . . 10  |-  ( O : dom  O -onto-> A  ->  ran  O  =  A )
5248, 49, 50, 514syl 21 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  =  A )
5312, 52eleqtrrd 2543 . . . . . . . 8  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  ran  O )
5453expr 615 . . . . . . 7  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  b  e.  ran  O ) )
5554pm2.18d 111 . . . . . 6  |-  ( (
ph  /\  b  e.  A )  ->  b  e.  ran  O )
5655ex 434 . . . . 5  |-  ( ph  ->  ( b  e.  A  ->  b  e.  ran  O
) )
5756ssrdv 3465 . . . 4  |-  ( ph  ->  A  C_  ran  O )
5811, 57eqssd 3476 . . 3  |-  ( ph  ->  ran  O  =  A )
59 isoeq5 6118 . . 3  |-  ( ran 
O  =  A  -> 
( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
6058, 59syl 16 . 2  |-  ( ph  ->  ( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
618, 60mpbid 210 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797   {crab 2800   _Vcvv 3072    i^i cin 3430    C_ wss 3431   class class class wbr 4395    |-> cmpt 4453    _E cep 4733   Se wse 4780    We wwe 4781   Oncon0 4822   dom cdm 4943   ran crn 4944   "cima 4946   Fun wfun 5515    Fn wfn 5516   -->wf 5517   -1-1->wf1 5518   -onto->wfo 5519   -1-1-onto->wf1o 5520   ` cfv 5521    Isom wiso 5522   iota_crio 6155  recscrecs 6936  OrdIsocoi 7829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-recs 6937  df-oi 7830
This theorem is referenced by:  ordtype  7852
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